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Topic: A formal counter-example of Ax Ey P(x,y) -> Ey Ax P(x,y)
Replies: 4   Last Post: Dec 9, 2012 12:39 AM

 Messages: [ Previous | Next ]
 Graham Cooper Posts: 4,495 Registered: 5/20/10
Re: A formal counter-example of Ax Ey P(x,y) -> Ey Ax P(x,y)
Posted: Dec 8, 2012 6:30 PM

On Dec 9, 9:05 am, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:
> On Dec 8, 1:58 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
>

> > On Dec 8, 6:14 am, Dan Christensen <d...@dcproof.com> wrote:
>
> > > Let the domain of quantification be U = {x, y} for distinct x and y.
>
> > > Let P be the "is equal to" relation on U.
>
> > > Then Ax Ey P(x,y) would be true since x=x and y=y
>
> > > And Ey Ax P(x,y) would be false since no element of U would be equal
> > > to every element of U.

>
> > > See formal proof (in DC Proof 2.0 format) athttp://dcproof.com/PopSci.htm
>
> > This is a classic Skolem Function example.
>
> This problem is central to predicate calculus. Like Russell's Paradox,
> it has spurred various "solutions," Skolem functions being one of
> them. My own DC Proof system is another, more natural one (IMHO).
>
> Dan

a fine grain solution, but I think relational models, set at a time
reasoning, although there is an intuitive understanding step required,
will be more amenable to formal proofs being so much shorter and in
the range of brute force proof search. 65 steps === n^65 is too big
for a computer to ever come across, and that was a simple quantifier
example.

Herc

Date Subject Author
12/8/12 Graham Cooper
12/8/12 Dan Christensen
12/8/12 Graham Cooper
12/9/12 Dan Christensen